If you can get away with it, keep all fruit out of the fridge. There are exceptions, but often the taste of fruit declines in the cold.

Apples will stay good a very long time in the fridge, with little or no drop in taste.

Watermelon, once you crack them open you should put the remainder in the fridge. Chilled watermelon is delicious.

Never put green bananas in the fridge, but once they're very ripe you can likely get another week out of them in a refrigerator. The skin may turn black in the cold, but the flesh will still be fine, with little taste deterioration.

I've found that cantaloupes decline after more than a day or two in the fridge.

Tomatoes should never be put in the fridge They'll last a week or eight days on the counter after getting red, and really shouldn't be kept longer than that. The tart taste of tomatoes is due to a chemical called Linolenic Acid converting to Z-3-Hexenel, and this reaction is disrupted by cold, so keep them at room temperature.

The All new Joy of Cooking (1997) says:
"After ripening, most fruits keep best when stored in a perforated plastic bag in the refrigerator crisper drawer. The exceptions - avocados, bananas, citrus fruit, pineapples, and melons - can be refrigerated, but their quality is best preserved in a dark, cool (50-65° F) place."

Stone fruit:
Store peaches and nectarines at room temperature until ripe - this usually takes 2 to 3 days. A ripe fruit will yield a bit when pressed gently. To speed up the ripening process, place nectarines or peaches in a paper bag and store at room temperature, out of direct sunlight. Storing ripe stone fruit in the crisper drawer will prolong its eating life - it should keep for up to a week when refrigerated. FreshDirect.com

Peaches:
Since peaches are highly perishable, purchase only the amount you need to use within a few days or plan on preserving them for long-term storage. The fruit should not be washed until just before you intend to use it. When storing fresh peaches at room temperature, be sure there is enough space between them to allow proper air circulation. Fresh peaches can be kept at room temperature three to four days, depending on how ripe they are when you buy them.

Refrigeration will extend their life, but not by more than a day or two. Peaches need humidity, so refrigerate in a plastic bag and use within two days. Ideally, let refrigerated peaches come close to room temperature before eating (about 30 minutes), as the flavor will be more full and robust.
Source: http://homecooking.about.com/od/foodstorage/a/peachstorage.htm

Cherries:
Since cherries don't ripen after harvest and tend to spoil quickly, they should be stored in the refrigerator, where they will keep for several days. They tend to absorb the odors of other foods, so keep them stored separately. Wash cherries just before serving.